PYQ 2019 - VedPrep

IIT JAM MATHEMATICS - 2019
Previous Year Question Paper with Solution.

1. Let a₁ = b₁ = 0, and for each n > 2, let a_n and b_n be real numbers given by

a_n = .

Then which one of the following is TRUE about the sequence {a_n} and {b_n}?

(a) Both {a_n} and {b_n} are divergent.

(b) {a_n} is convergent and {b_n} is divergent.

(d) Both {a_n} and {b_n} are convergent.

Ans. (d)

Sol.

2. Let Let V be the subspace of defined by

Then the dimension of V is

(a) pn – rank(T)

(b) mn – p rank(T)

(d) p(n – rank(T))

Ans. (d)

Sol.

We know that every matrix represent a linear transformation

3. Let be a twice differentiable function. Define by

f(x, y, z) = g(x² + y² – 2z²).

Then is equal to

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

4. Let be sequences of positive real numbers such that na_n < b_n < n²a_n for all n >
2. If the radius of convergence of the power series then the power series

(a) converges for all x with |x| < 2

(b) converges for all x with |x| > 2

(d) does not converge for any x with |x| < 2

Ans. (a)

Sol. Let be sequence of positive real numbers such that

Given, Power series = and radius of convergence = 4.

5. Let S be the set of all limit points of the set . Let Q₊ be the set of all positive rational numbers. Then

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Let

6. If x^hy^k is an integrating factor of the differential equation y(1 + xy)dx + x(1 – xy)dy = 0 then the ordered pair (h, k) is equal to

(a) (–2, –2)

(b) (–2, –1)

(d) (–1, –1)

Ans. (a)

Sol. Given differential equation y(1 + xy)dx + x(1 – xy) = 0 .

This equation is in the form of

Given, that integrating factor is in the form of xⁿy^k, so by comparing

h = –2, k = –2

So, (h, k) = (–2, –2).

7. If y(x) = is a solution of the differential equation

satisfying (0) = 5 then y(0) is equal to

(a) 1

(b) 4

(d) 9

Ans. (c)

Sol. Given, differential equation

Characteristic equation of given differential equation is

8. The equation of the tangent plane to the surface x²z + = 6 at the point (2, 0, 1) is

(a) 2x + z = 5

(b) 3x + 4z = 10

(d) 7x – 4z = 10

Ans. (b)

Sol. Let equation of tangent plane

9. The value of integral is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Let

Here, upper limit x = 1 – y²

Change the order of integral

10. The area of the surface generated by rotating the curve x = y³, about the y-axis is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We have, curve x = y³

The area of the surface generated by rotating the curve is

11. Let H and K be subgroups of If the order of H is 24 and the order of K is 36 then the order of subgroup is

(a) 3

(b) 4

(d) 12

Ans. (d)

Sol. Let H and K be a subgroup of Z₁₄₄if order of H is 24 and the order of K is 36.

12. Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with i = is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix then

(a) P⁴ = 4P² + 9I

(b) P⁴ = 4P² – 9I

(d) P⁴ = 2P² + 9I

Ans. (c)

Sol.

13. The set as a subset of is

(a) connected and compact

(b) connected but not compact

(d) neither connected nor compact

Ans. (b)

Sol. Let

So, function is increasing, then maximum value at x = 1 and minimum at x = –1

14. The set as a subset of R is

(a) compact and open

(b) compact but not open

(d) neither compact nor open

Ans. (b)

Sol. The set

Limit point of

All limit points of set is in the set.

So, A is closed.

Set is bounded, So, set is compact.

Hence, set is compact but not open.

15. For –1 < x < 1, the sum of the power series 1 + is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

Differential w.r.t. 'x'

16. Let f(x) = (ln x)², x > 0. Then

(a)

(b)

(c)

(d) does not exist

Ans. (c)

Sol. Let we have f(x) = (log x)²

Now, option (c)

Using L'Hospital Rule

Again, by L'Hospital Rule

by Hospital Rule

17. Let be a differentiable function such that and f(0) = 1. Then f(1) lies in the interval

(a) (0, e^–1)

(b)

(c)

(d)

Ans. (d)

Sol. Given that

Integrating both sides with limit 0 to 1.

18. For which one of the following values of k, the equation 2x³ + 3x² – 12x – k = 0 has three distinct real roots?

(a) 16

(b) 20

(d) 31

Ans. (a)

Sol. Given equation,

2x³ + 3x² – 12x – k = 0

⇒ 2x³ + 3x² – 12x = k

For three real roots.

Let f(x) = 2x³ + 3x² – 12x = x(2x² + 3x – 12)

Now, roots are

Now, for minima and maxima in

19. Which one of the following series is divergent?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

20. Let S be the family of orthogonal trajectories of the family of curves

If passes through the point (1, 2), then C also passes through

(a)

(b) (2, –4)

(c)

(d)

Ans. (c)

Sol. The family of curves

Now, for orthogonal trajectories

21. Let x, x + e^x and 1 + x + e^x be solutions of a linear second order ordinary differential equation with constant coefficients. If y(x) is the solution of the same equation satisfying y(0) = 3 and y(0) = 4 then y(1) is equal to

(a) e + 1

(b) 2e + 3

(d) 3e + 1

Ans. (d)

Sol. Let a linear second order ordinary diff. equation with constant coefficient is

General solution is

Given that, x, x + e^x and 1 + x + e^x be the solution of given equation

So, P(x) = x, y₁ = 1 = e^0x and y₂ = e^x then

22. The function f(x, y) = x³ + 2xy + y³has a saddle point at

(a) (0, 0)

(b)

(c)

(d) (–1, –1)

Ans. (a)

Sol. We have

Substitute in Eq. (ii), we get

23. The area of the part of the surface of the paraboloid x² + y² + z = 8 lying inside the cylinder x² + y² = 4 is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The area of a surface, f(x, y), given a region R of the XY-plane is given by

where are the partial derivatives of f(x, y) with respect to x and y respectively.

Hence, the surface area S is given by

S =

In converting the integral of a function in rectangular coordinates to a function in polar coordinates:

24. Let C be the circle (x – 1)² + y² = 1 oriented counter clockwise. Then the value of the line integral

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Equation of circle

(x – 1)² + y² = 1 with parametric equation x = 1 + cos t, y = sin t

dx = –sin tdt and dy = cos t dt

We know that

25. Let and let C
be the curve of intersection of the plane x + y + z = 1 and cylinder x² + y² = 1. Then the value of is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

26. The tangent line to the curve of intersection of the surfaces x² + y² – z = 0 and the plane x + z = 3 at the point (1, 1, 2) passes through

(a) (–1, –2, 4)

(b) (–1, 4, 4)

(d) (–1, 4, 0)

Ans. (b)

Sol. We have

Now, equation of line passes through (1, 1, 2) and direction vector is parallel to is

By checking options (–1, 4, 4) substitute in equation

Hence, point (–1, 4, 4) satisfy the equation of tangent.

27. The set of eigen values of which one of the following matrices is NOT equal to the set of eigenvalues of ?

(a)

(b)

(c)

(d)

Ans.

Sol. Let A =

Characteristic polynomial of A

By checking options

So, eigenvalues of D is not equal to eigen values of given matrix.

28. Let {a_n} be a sequence of positive real numbers. The series converges if the series

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

So, option (c) is correct.

29.

Then at (0, 0), the function is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

Let y = mx², find limit at particular y = mx²

30. Let {a_n} be a sequence of positive real numbers such that

Then the sum of the series lies in the interval

(a) (1, 2]

(b) (2, 3]

(d) (4, 5]

Ans. (a)

Sol. Let {a_n} be a sequence of positive real numbers such that

31. Let G be a non-cyclic group of order 4. Consider the statements I and II:

I. There is NOT injective (one-one) homomorphism from G to ' ₈

II. There is NOT surjective (onto) homomorphism from ' ₈ to G

Then

(a) I is true

(b) I is false

(d) II is false

Ans. (a), (c)

Sol. Let G be a non-cyclic group of order 4.

(i) In homomorphism from G to Z₈

|G| = 4, |Z₈| = 8

in domain, there is 4 element and in co-domain 8 element. So, one-one mapping is not possible.

So, statement I is true.

(ii) Homomorphism from Z₈ to G, G be a non-cyclic group,

We know, that non-cyclic group is Klan 4-group.

Here, Z₈
G. There is no onto homomorphism shows from Z₈ to G.

Hence, statement II is true.

32. Let G be a non-abelian group and let the maps f, g, h from G to itself be defined by

f(x) = yxy^–1, g(x) = x^–1 and h = .

Then

(a) g and h are homomorphisms and f is not a homomorphism

(b) h is a homomorphism and g is not a homomorphism

(d) f, g and h are homomorphisms

Ans. (b), (c)

Sol. We have

So, h is an identity transformation and identity mapping is homomorphism.

Hence, h is homomorphism.

33. Let S and T be linear transformations from a finite dimensional vector space V to itself such that S(T(v)) = 0 for all v ∈ V. Then

(a)

(b)

(c)

(d)

Ans. (c), (d)

Sol. Let S and T be linear transformation from a finite dimensional vector space V to itself

34. Let be differentiable vector fields and let g be a differentiable scalar function. Then

(a)

(b)

(c)

(d)

Ans. (a), (d)

Sol. If F and G be differential vector field and let g be a differential scalar function. Then,

Now, RHS

Similarly,

Now, Eq. (i) – (ii), we get

And again

Now,

35. Consider the intervals S = (0, 2] and T = [1, 3). Let S° and T° be the sets of interior points of S and T respectively. Then the set of interior points of S\T is equal to

(a) S\T°

(b) S\T

(d) S°\T

Ans. (b), (d)

Sol. We have, the interval S = (0, 2] and T = [1, 3]

Let T⁰ and S⁰ be the sets of interior of T and S respectively.

S⁰ = (0, 2), T⁰ = (1, 3)

Now, S/T = (0, 1)

So, (S/T)⁰ = S/T = (0, 1)

and S⁰/T = (0, 1) = S/T

Hence, option (b) and (d) are correct.

36. Let {a_n} be the sequence given by

Then which of the following statements is/are TRUE about the subsequences {a_{6n – 1}} and {a_{6n + 4}}?

(a) Both the subsequences are convergent

(b) Only one of the subsequences is convergent

(c)

(d)

Ans. (a)

Sol. We have

Hence, both the subsequence are convergent.

37. Let

If h(x) = f(g(x)) then

(a) h is not differentiable at x = 0

(b)

(c)

(d)

Ans. (b), (c), (d)

Sol.

Now,

Now, again

38.

Then which of the following statements is/are TRUE?

(a) f is an increasing function

(b) f is a decreasing function

(c)

(d)

Ans. (b), (c)

Sol. We have

39. Let f(x, y) =

Then at (0, 0).

(a) f is continuous

(b)

(c)

(d)

Ans. (a), (d)

Sol. We have

f(x, y) =

So, at (0, 0), f is continuous.

Now,

Again

Hence, option (a) and (d) is correct.

40. Let {a_n} be the sequence of real number such that

Then

(a) a₄ = a₁(1 + a₁)(1 + a₂)(1 + a₃)

(b)

(c)

(d)

Ans. (a), (b)

Sol. We have

By checking options

So, option (a) is correct.

Hence, option (b) is correct.

41. Let x be 100-cycle (1 2 3 ... 100) and let y be the transposition (49 50) in the permutation group S₁₀₀. Then the order of xy is _________.

Ans. 99

Sol. Let x be the 100-cycle (1, 2 ... 100) and y be the transposition (49, 50) in the permutation group S₁₀₀.

xy = (1, 2, 3 ... 49, 50, 51 ... 100) (49, 50)

= (1, 2, 3 ... 48, 51, 52, ... 100) (50)

in xy, the element 50 is not available.

xy is cycle of 99.

Hence, order of xy is 99.

42. Let W₁ and W₂ be subspaces of the real vector space defined by

W₁ = {(x₁, x₂, ..., x₁₀₀) : x_i = 0 if i is divisible by 4}

W₂ = {(x₁, x₂, ..., x₁₀₀) : x_i = 0 if i is divisible by 5}

Then the dimension of is _________.

Ans. 60

Sol. We have,

W₁ = {(x₁, x₂ ... x₁₀₀) : x₁ = 0, if i is divisible by 4}

Hence, x₄, x₈ ... x₁₀₀ are equal to zero. So total number of zeros = 40.

Again,

W₂ = {(x₁, x₂ ... x₁₀₀) : x_i = 0, if i is divided by 5}

Hence x₅, x₁₀, x₁₅, ..., x₁₀₀ are equal to zero. So total no. of zeros = 20.

Now, LCM of 4 and 5 is 20 and multiple of 20 is 20, 40, 60, 80 and 100.

So, = 100 – 25 – 20 + 5 = 60.

43. Consider the following system of three linear equations in four unknowns x₁, x₂, x₃ and x₄

x₁ + x₂ + x₃ + x₄ = 4

x₁ + 2x₂ + 3x₃ + 4x₄ = 5,

x₁ + 3x₂ + 5x₃ + kx₄ = 5.

If the system has no solutions, then k = _________.

Ans. 7

Sol. System of three linear equations in four unknowns. x₁, x₂, x₃, x₄ are

x₁ + x₂ + x₃ + x₄ = 4

x₁ + 2x₂ + 3x₃ + 4x₄ = 5

x₁ + 3x₂ + 5x₃ + kx₄ = 5

argumented matrix is

44. Let be the ellipse

oriented counter clockwise. Then the value of (round off to 2 decimal places) is _________.

Ans. 75.36

Sol. We have

45. The coefficient of in the Taylor series expansion of the function

f(x) =

about x = is _________.

Ans. (1)

Sol. We have

f(x) =

46.

f(x) = .

Then is _________.

Ans. 0.5

Sol.

47. If g(x) =

then

Ans. 2.6

Sol.

48. Let f(x, y) =

Then the directional derivative of f at (0, 0) in the direction of is _________.

Ans. 8

Sol. We have

49. The value of the integral (round off to 2 decimal places) is _________.

Ans. 2.67

Sol. We have

We know that

Here, in graph region I is equal to region II.

So,

= 2.66 = 2.67 (round off to 2 decimal).

50. The volume of the solid bounded by the surfaces x = 1 – y² and x = y² – 1 and the planes z = 0 and z = 2 (round off to 2 decimal places) is _________.

Ans. 5.30 to 5.50

Sol.

51. The volume of the solid of revolution of the loop of the curve y² = x⁴(x + 2) about the x-axis (round off to 2 decimal places) is _________.

Ans. 6.70

Sol. The given curve is y² = x⁴ (x + 2) .

When y = 0 the values of x is 0 and –2 .

52. The greatest lower bound of the set (round off to 2 decimal places) is _________.

Ans. 2.71

Sol. Let

Here, a_n is monotonically decreasing sequence, then it converges to its greatest lower bound.

So, greatest lower bound = e = 2.71.

53. Let G = be the group under multiplication modulo 55. Let be such that x² = 26 and x > 30. Then x is equal to _________.

Ans. 31, 46

Sol.

Be group under multiplication modulo 55 be such that x² = 26 and x > 30

Now, verify for

54. The number of critical points of the function f(x, y) = is _________.

Ans. 5

Sol.

For critical points

From eqs. (i) and (ii)

55. The number of elements in the set where e is the identity element of the permutation group S₃ is _________.

Ans. 4

Sol. We have where e is the identity element we know that

S₃ = {e, (12), (13), (2 3), (1 3 2), (1 2 3)}

Now, x⁴ = e = e⁴ = e

and (1 2), (1 3), (2 3) are element of order 2.

⇒ x² = e

⇒ x⁴ = x² – x² = e – e = e

Hence, there are 4 elements in the set .

56. If is an eigenvector corresponding to a real eigenvalue of the matrix then z – y is equal to _________.

Ans. 3

Sol. Given matrix A =

The characteristic polynomial is

57. Let M and N be any two 4 × 4 matrices with integer entries satisfying

MN = .

Then the maximum value of det(M) + det(N) is _________.

Ans. 17

Sol. Given MN =

det (MN) = det (M) · det(N) = 2⁴ × 1

det (MN) = 16.

Here, M and N are matrices with integer entries, so possible values of M and N is factors of 16.

So, maximum values of det(M) + det(N) = 16 + 1 = 17.

58. Let M be 3 × 3 matrix with real entries such that M² = M + 2I where I denotes the 3 × 3 identity matrix. are eigenvalues of M such that is equal to _________.